The positive part of Onsager's conjecture was proved by Constantin-E-Titi in [1]},
where they showed that energy is conserved for a weak solution \(v\)  in the Besov space \(L^{3}(0,T; B^{\alpha }_{3,\infty }(\mathbb {T}^{3}))\)  with \(\alpha >1/3\) . For \(\alpha =1/3\) , it is shown that every weak solution \(v\)  of the Euler equations
on \([0, T ]\)  conserves energy provided \(v \in L^{3}(0,T; B^{1/3}_{3,c(N)})\)  by
Cheskidov-Constantin-Friedlander-Shvydkoy in [2]}, where
\(B^{1/3}_{3,c(N)}=\lbrace (u\in B^{1/3}_{3,\infty }, \lim _{q\rightarrow \infty }\int _{0}^{T}2^{q}\Vert \Delta _{q}u\Vert ^{3}_{L^{3}}dt=0)\rbrace \) .
After a series of papers by De Lellis and Szekelyhidi [3]}, [4]}, [5]}, [6]}, Isett [7]} successfully solved the second part of Onsager's conjecture in three-dimensional space.
At this stage, the progress of Onsager's conjecture of the 3D incompressible Euler equations (REF ) is
satisfactory.
